Introduction to quantum effects in gravity:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2007
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | X, 273 S. Ill., graph. Darst. |
| ISBN: | 9780521868341 |
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| 245 | 1 | 0 | |a Introduction to quantum effects in gravity |c Viatcheslav Mukhanov and Sergei Winitzki |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2007 | |
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Record in the Search Index
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| adam_text | Contents
Preface page ix
Part I Canonical quantization and particle production 1
1 Overview: a taste of quantum fields 3
1.1 Classical field 3
1.2 Quantum field and its vacuum state 4
1.3 The vacuum energy 7
1.4 Quantum vacuum fluctuations 8
1.5 Particle interpretation of quantum fields 9
1.6 Quantum field theory in classical backgrounds 9
1.7 Examples of particle creation 10
2 Reminder: classical and quantum theory 13
2.1 Lagrangian formalism 13
2.1.1 Functional derivatives 15
2.2 Hamiltonian formalism 17
2.3 Quantization of Hamiltonian systems 19
2.4 Hilbert spaces and Dirac notation 20
2.5 Operators, eigenvalue problem and basis in a •_•• ¦ ¦¦¦
Hilbert space ¦ ^ ; 22
2.6 Generalized eigenvectors and basic matrix elements , 26
2.7 Evolution in quantum theory . ¦ • 29
K ¦ *
3 Driven harmonic oscillator 33
3.1 Quantizing an oscillator • 33
3.2 The in and out states 35
3.3 Matrix elements and Green s functions 38
V
vi Contents
4 From harmonic oscillators to fields 42
4.1 Quantum harmonic oscillators 42
4.2 From oscillators to fields 43
4.3 Quantizing fields in a flat spacetime 45
4.4 The mode expansion 48
4.5 Vacuum energy and vacuum fluctuations 50
4.6 The Schrodinger equation for a quantum field 51
5 Reminder: classical fields 54
5.1 The action functional 54
5.2 Real scalar field and its coupling to the gravity 56
5.3 Gauge in variance and coupling to the electromagnetic field 58
5.4 Action for the gravitational and gauge fields 59
5.5 Energy momentum tensor 61
6 Quantum fields in expanding universe 64
6.1 Classical scalar field in expanding background 64
6.1.1 Mode expansion 66
6.2 Quantization 67
6.3 Bogolyubov transformations 68
6.4 Hilbert space; a and b particles 69
6.5 Choice of the physical vacuum 71
6.5.1 The instantaneous lowest energy state 71
6.5.2 Ambiguity of the vacuum state 74
6.6 Amplitude of quantum fluctuations 78
6.6.1 Comparing fluctuations in the vacuum and excited
states 80
6.7 An example of particle production 81
7 Quantum fields in the de Sitter universe 85
7.1 De Sitter universe 85
7.2 Quantization 88
7.2.1 Bunch Davies vacuum 90
7.3 Fluctuations in inflationary universe 92
8 Unruh effect 97
8.1 Accelerated motion 97
8.2 Comoving frame of accelerated observer 100
8.3 Quantum fields in inertial and accelerated frames 103
8.4 Bogolyubov transformations 106
8.5 Occupation numbers and Unruh temperature 107
Contents vii
9 Hawking effect. Thermodynamics of black holes 109
9.1 Hawking radiation 109
9.1.1 Schwarzschild solution 110
9.1.2 Kruskal—Szekeres coordinates 111
9.1.3 Field quantization and Hawking radiation 115
9.1.4 Hawking effect in 3 + 1 dimensions 118
9.2 Thermodynamics of black holes 120
9.2.1 Laws of black hole thermodynamics 121
10 The Casimir effect 124
10.1 Vacuum energy between plates 124
10.2 Regularization and renormalization 125
Part II Path integrals and vacuum polarization 129
11 Path integrals 131
11.1 Evolution operator. Propagator 131
11.2 Propagator as a path integral 132
11.3 Lagrangian path integrals 137
11.4 Propagators for free particle and harmonic oscillator 138
11.4.1 Free particle 139
11.4.2 Quadratic potential 139
11.4.3 Euclidean path integral 142
11.4.4 Ground state as a path integral 144
12 Effective action 146
12.1 Driven harmonic oscillator (continuation) 146
12.1.1 Green s functions and matrix elements 146
12.1.2 Euclidean Green s function 148
12.1.3 Introducing effective action 150
12.1.4 Calculating effective action for a driven oscillator 152
12.1.5 Matrix elements 154
12.1.6 The effective action recipe 157
12.1.7 Backreaction 158
12.2 Effective action in external gravitational field 159
12.2.1 Euclidean action for scalar field 161
12.3 Effective action as a functional determinant 163
12.3.1 Reformulation of the eigenvalue problem 164
12.3.2 Zeta function 166
12.3.3 Heat kernel 167
viii Contents
13 Calculation of heat kernel 170
13.1 Perturbative expansion for the heat kernel 171
13.1.1 Matrix elements 172
13.2 Trace of the heat kernel 176
13.3 The Seeley DeWitt expansion 178
14 Results from effective action 180
14.1 Renormalization of the effective action 180
14.2 Finite terms in the effective action 183
14.2.1 EMT from the Polyakov action 185
14.3 Conformal anomaly 187
Appendix 1 Mathematical supplement 193
A 1.1 Functionals and distributions (generalized
functions) 193
A1.2 Green s functions, boundary conditions, and
contours 202
A 1.3 Euler s gamma function and analytic
continuations 206
Appendix 2 Backreaction derived from effective action 212
Appendix 3 Mode expansions cheat sheet 216
Appendix 4 Solutions to exercises 218
Index 272
|
| adam_txt |
Contents
Preface page ix
Part I Canonical quantization and particle production 1
1 Overview: a taste of quantum fields 3
1.1 Classical field 3
1.2 Quantum field and its vacuum state 4
1.3 The vacuum energy 7
1.4 Quantum vacuum fluctuations 8
1.5 Particle interpretation of quantum fields 9
1.6 Quantum field theory in classical backgrounds 9
1.7 Examples of particle creation 10
2 Reminder: classical and quantum theory 13
2.1 Lagrangian formalism 13
2.1.1 Functional derivatives 15
2.2 Hamiltonian formalism 17
2.3 Quantization of Hamiltonian systems 19
2.4 Hilbert spaces and Dirac notation 20
2.5 Operators, eigenvalue problem and basis in a •_•• ¦ ¦¦¦
Hilbert space ¦ ^ ; 22
2.6 Generalized eigenvectors and basic matrix elements , 26
2.7 Evolution in quantum theory \ . ¦ •' 29
K ¦ *
3 Driven harmonic oscillator \ \ 33
3.1 Quantizing an oscillator • 33
3.2 The "in" and "out" states 35
3.3 Matrix elements and Green's functions 38
V
vi Contents
4 From harmonic oscillators to fields 42
4.1 Quantum harmonic oscillators 42
4.2 From oscillators to fields 43
4.3 Quantizing fields in a flat spacetime 45
4.4 The mode expansion 48
4.5 Vacuum energy and vacuum fluctuations 50
4.6 The Schrodinger equation for a quantum field 51
5 Reminder: classical fields 54
5.1 The action functional 54
5.2 Real scalar field and its coupling to the gravity 56
5.3 Gauge in variance and coupling to the electromagnetic field 58
5.4 Action for the gravitational and gauge fields 59
5.5 Energy momentum tensor 61
6 Quantum fields in expanding universe 64
6.1 Classical scalar field in expanding background 64
6.1.1 Mode expansion 66
6.2 Quantization 67
6.3 Bogolyubov transformations 68
6.4 Hilbert space; "a and b particles" 69
6.5 Choice of the physical vacuum 71
6.5.1 The instantaneous lowest energy state 71
6.5.2 Ambiguity of the vacuum state 74
6.6 Amplitude of quantum fluctuations 78
6.6.1 Comparing fluctuations in the vacuum and excited
states 80
6.7 An example of particle production 81
7 Quantum fields in the de Sitter universe 85
7.1 De Sitter universe 85
7.2 Quantization 88
7.2.1 Bunch Davies vacuum 90
7.3 Fluctuations in inflationary universe 92
8 Unruh effect 97
8.1 Accelerated motion 97
8.2 Comoving frame of accelerated observer 100
8.3 Quantum fields in inertial and accelerated frames 103
8.4 Bogolyubov transformations 106
8.5 Occupation numbers and Unruh temperature 107
Contents vii
9 Hawking effect. Thermodynamics of black holes 109
9.1 Hawking radiation 109
9.1.1 Schwarzschild solution 110
9.1.2 Kruskal—Szekeres coordinates 111
9.1.3 Field quantization and Hawking radiation 115
9.1.4 Hawking effect in 3 + 1 dimensions 118
9.2 Thermodynamics of black holes 120
9.2.1 Laws of black hole thermodynamics 121
10 The Casimir effect 124
10.1 Vacuum energy between plates 124
10.2 Regularization and renormalization 125
Part II Path integrals and vacuum polarization 129
11 Path integrals 131
11.1 Evolution operator. Propagator 131
11.2 Propagator as a path integral 132
11.3 Lagrangian path integrals 137
11.4 Propagators for free particle and harmonic oscillator 138
11.4.1 Free particle 139
11.4.2 Quadratic potential 139
11.4.3 Euclidean path integral 142
11.4.4 Ground state as a path integral 144
12 Effective action 146
12.1 Driven harmonic oscillator (continuation) 146
12.1.1 Green's functions and matrix elements 146
12.1.2 Euclidean Green's function 148
12.1.3 Introducing effective action 150
12.1.4 Calculating effective action for a driven oscillator 152
12.1.5 Matrix elements 154
12.1.6 The effective action "recipe" 157
12.1.7 Backreaction 158
12.2 Effective action in external gravitational field 159
12.2.1 Euclidean action for scalar field 161
12.3 Effective action as a functional determinant 163
12.3.1 Reformulation of the eigenvalue problem 164
12.3.2 Zeta function 166
12.3.3 Heat kernel 167
viii Contents
13 Calculation of heat kernel 170
13.1 Perturbative expansion for the heat kernel 171
13.1.1 Matrix elements 172
13.2 Trace of the heat kernel 176
13.3 The Seeley DeWitt expansion 178
14 Results from effective action 180
14.1 Renormalization of the effective action 180
14.2 Finite terms in the effective action 183
14.2.1 EMT from the Polyakov action 185
14.3 Conformal anomaly 187
Appendix 1 Mathematical supplement 193
A 1.1 Functionals and distributions (generalized
functions) 193
A1.2 Green's functions, boundary conditions, and
contours 202
A 1.3 Euler's gamma function and analytic
continuations 206
Appendix 2 Backreaction derived from effective action 212
Appendix 3 Mode expansions cheat sheet 216
Appendix 4 Solutions to exercises 218
Index 272 |
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| spelling | Mukhanov, Viatcheslav F. Verfasser (DE-588)131984047 aut Introduction to quantum effects in gravity Viatcheslav Mukhanov and Sergei Winitzki 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2007 X, 273 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Hier auch später erschienene, unveränderte Nachdrucke Gravity Quantum field theory Quantum gravity Gravitation (DE-588)4021908-2 gnd rswk-swf Quantenfeldtheorie (DE-588)4047984-5 gnd rswk-swf Quantenfeldtheorie (DE-588)4047984-5 s Gravitation (DE-588)4021908-2 s DE-604 Winitzki, Sergei Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015710806&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mukhanov, Viatcheslav F. Winitzki, Sergei Introduction to quantum effects in gravity Gravity Quantum field theory Quantum gravity Gravitation (DE-588)4021908-2 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd |
| subject_GND | (DE-588)4021908-2 (DE-588)4047984-5 |
| title | Introduction to quantum effects in gravity |
| title_auth | Introduction to quantum effects in gravity |
| title_exact_search | Introduction to quantum effects in gravity |
| title_exact_search_txtP | Introduction to quantum effects in gravity |
| title_full | Introduction to quantum effects in gravity Viatcheslav Mukhanov and Sergei Winitzki |
| title_fullStr | Introduction to quantum effects in gravity Viatcheslav Mukhanov and Sergei Winitzki |
| title_full_unstemmed | Introduction to quantum effects in gravity Viatcheslav Mukhanov and Sergei Winitzki |
| title_short | Introduction to quantum effects in gravity |
| title_sort | introduction to quantum effects in gravity |
| topic | Gravity Quantum field theory Quantum gravity Gravitation (DE-588)4021908-2 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd |
| topic_facet | Gravity Quantum field theory Quantum gravity Gravitation Quantenfeldtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015710806&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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